This past weekend I got Rota's 1996 book Indiscrete Thoughts from the
library and read chapters 1 through 15, more than half of the book.
Surprisingly, I found it congenial. Why surprisingly? Because Rota's
book has been cited approvingly by Reuben Hersh as supporting Hersh's
"humanist" (i.e. social constructivist) agenda.
Actually, Rota's ideas have no connection to Hersh's that I can
discern. There is none of the pointless hostility to f.o.m., none of
the fuzzy sociology, none of the absurd political calculus, none of
the incoherent generalities, none of the bombast. Instead Rota
provides an acute, honest view of mathematics and mathematicians, plus
some original and sometimes intriguing philosophical speculations.
The chapters of Rota's book are self-contained short essays which I
imagine have been published separately elsewhere.
Chapters 1 and 2 discuss some mathematicians that Rota met as a
student at Princeton and Yale -- Church, Feller, Artin, Lefschetz,
Dunford, Schwartz -- warts and all. Chapters 4 through 6 present a
controversial and unflattering portrait of Rota's friend Ulam.
Chapter 3 is an enlightening historical discussion of certain topics
in algebra whose "bottom line" is combinatorial (Young tableaux,
Grassmann exterior algebras, ..), with an interesting contrast to the
more mainstream or fashionable or perhaps snooty kind of algebra whose
"bottom line" is algebraic number theory and algebraic geometry. Rota
refers to these as Algebra Two and Algebra One, respectively. Shades
of set-theory-1 and set-theory-2? (See FOM 17 Sep 1998 12:37:54.)
Chapters 18 through 21 are a grab-bag of gossip, book reviews, advice
for young mathematicians, etc. I may report more on these chapters
after I have read them through. I hope they will not disgust me.
There is reason to believe that they *will* disgust me, because some
of Rota's ideas and practices regarding evaluation of mathematical
writing and research are known to be cynical in the extreme. However,
I intend to keep an open mind.
Chapters 7 through 17 present Rota's ideas about philosophy of
mathematics and phenomenology. These are for me the most interesting
part of the book.
Chapter 7 ("The Pernicious Influence of Mathematics Upon Philosophy")
is a withering attack on analytical philosophy, withering because
written by a mathematician who can speak with some authority on the
limitations of formal methods. [ Actually, philosophers themselves
have written books which present a far more devastating critique of
analytical philosophy. My favorite is Reason and Analysis, by Brand
Blanshard. ]
Chapter 8 is a mysterious (to me) call for artificial intelligence
researchers to heed some distinctions drawn by 20th century
philosophers including Wittgenstein.
Chapters 9, 10, 11, and 14 present Rota's phenomenological perspective
on mathematical truth, beauty and proof. I'm not sure how much this
has to do with Husserl and Heidegger, but it certainly contains some
good insights. I especially like Rota's explanation of mathematical
beauty -- it boils down to enlightenment. I also like Rota's idea
that there is need for *more* rigor, specifically a more rigorous
treatment of notions such as understandability, simplicity, and
definitiveness of a proof.
Chapters 12 and 13 contain questionable ruminations on the fact that
there can be disparate axiomatizations of one and the same
mathematical concept. An excerpt:
"An adequate understanding of mathematical identity requires a
missing theory that will account for the relationships between
formal systems that describe the same items. At present, such
relationships can at best be heuristically described in terms that
invoke some notion of an intelligent user standing outside the
system."
Oddly, Rota does not mention formal interpretability as a candidate
for the missing theory.
Chapter 15 ("Fundierung as a Logical Concept") discusses the idea of a
material fact as representing a functionality. Examples: a pen as a
writing instrument, Rota as a teacher, Rota as a taxpayer, a dollar
amount as a price, the queen of hearts as a card in a bridge game.
Rota suggests that there is a need for a rigorous logical analysis of
`as' (or `qua', although Rota doesn't use this term).
I'm looking forward to chapters 16 and 17. Perhaps I'll report later.
-- Steve